论文标题
Coxeter组的船体度量
The hull metric on Coxeter groups
论文作者
论文摘要
我们重新解释了最初是由于Sidorenko的不等式，该poset的线性扩展是根据对称组的凸子集$ \ mathfrak {s} _n $的线性扩展。我们推测，类似的不平等现象以任意（不必要的）Coxeter组$ W $存在，并为Hyperoctahedral groups $ b_n $和所有权利角coxeter组证明了这一点。我们对$ b_n $的证明（以及$ \ mathfrak {s} _n $）使用组合插入图与线性扩展上的良好促销操作员密切相关的组合插入图；该地图可能具有独立的兴趣。我们还注意到，所讨论的不平等现象可以解释为三角形不等式，因此，只要我们的猜想持有，凸面就可以用来定义$ w $的新不变度度量。该指标的几何特性是未来研究的有趣方向。
We reinterpret an inequality, due originally to Sidorenko, for linear extensions of posets in terms of convex subsets of the symmetric group $\mathfrak{S}_n$. We conjecture that the analogous inequalities hold in arbitrary (not-necessarily-finite) Coxeter groups $W$, and prove this for the hyperoctahedral groups $B_n$ and all right-angled Coxeter groups. Our proof for $B_n$ (and new proof for $\mathfrak{S}_n$) use a combinatorial insertion map closely related to the well-studied promotion operator on linear extensions; this map may be of independent interest. We also note that the inequalities in question can be interpreted as a triangle inequalities, so that convex hulls can be used to define a new invariant metric on $W$ whenever our conjecture holds. Geometric properties of this metric are an interesting direction for future research.
